Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(680\)\(\medspace = 2^{3} \cdot 5 \cdot 17 \) |
| Artin field: | Galois closure of 4.0.39304000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{680}(13,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} + 170x^{2} + 3060 \)
|
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 22\cdot 23^{2} + 13\cdot 23^{3} + 20\cdot 23^{4} + 5\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 7\cdot 23 + 9\cdot 23^{2} + 9\cdot 23^{3} + 21\cdot 23^{4} + 6\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 15\cdot 23 + 13\cdot 23^{2} + 13\cdot 23^{3} + 23^{4} + 16\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 + 22\cdot 23 + 9\cdot 23^{3} + 2\cdot 23^{4} + 17\cdot 23^{5} +O(23^{6})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ | ✓ |
| $1$ | $4$ | $(1,2,4,3)$ | $\zeta_{4}$ | |
| $1$ | $4$ | $(1,3,4,2)$ | $-\zeta_{4}$ |